MEN OF MATHEMATICS Hhree*? An answer is contained in the following definition: 'The number of things hi a given class is the class of all classes that are similar to the given class.' This definition gains nothing from attempted explanation: it must be grasped as it is. It was proposed in 1879 by Gottlob Frege, and again (independently) by Bertrand Russell in 1901. One advantage which it has over other definitions of 'cardinal number of a class' is its applicability to both finite and infinite classes. Those who believe the definition too mystical for mathe- matics can avoid it by following Couturat's advice and not attempting to define 'cardinal number'. However, that way also leads to difficulties. Cantor's spectacular result that the class of all algebraic numbers is similar (hi the technical sense defined above) to its sub-class of all the positive rational integers was but the first of many wholly unexpected properties of infinite classes, Granting for the moment that his reasoning in reaching these properties is sound, or, if not unobjectionable in the form in which Cantor left it, that it can be made rigorous, we must admit its power. Consider for example the 'existence' of transcendental numbers. In an earlier chapter we saw what a tremendous effort it cost Hermite to prove the transcendence of a particular number of this kind. Even to-day there is no general method known whereby the transcendence of any number which we suspect is transcendental can be proved; each new type requires the invention of special and ingenious methods. It is suspected, for example, that the number (it is a constant, although it looks as if it might be a variable from its definition) which is defined as the limit of as n tends to infinity, is transcendental, but we cannot prove that it is. What is required is to show that this constant is not a root of any algebraic equation with rational integer co- efficients. All this suggests the question *How many transcendental 626